You have found the following ages (in years) of all 4 sloths at your local zoo: $ 2,\enspace 5,\enspace 20,\enspace 13$ What is the average age of the sloths at your zoo? What is the standard deviation? You may round your answers to the nearest tenth.
Explanation: Because we have data for all 4 sloths at the zoo, we are able to calculate the population mean $({\mu})$ and population standard deviation $({\sigma})$ To find the population mean , add up the values of all $4$ ages and divide by $4$ $ {\mu} = \dfrac{\sum\limits_{i=1}^{{N}} x_i}{{N}} = \dfrac{\sum\limits_{i=1}^{{4}} x_i}{{4}} $ $ {\mu} = \dfrac{2 + 5 + 20 + 13}{{4}} = {10\text{ years old}} $ Find the squared deviations from the mean for each sloth. Age $x_i$ Distance from the mean $(x_i - {\mu})$ $(x_i - {\mu})^2$ $2$ years $-8$ years $64$ years $^2$ $5$ years $-5$ years $25$ years $^2$ $20$ years $10$ years $100$ years $^2$ $13$ years $3$ years $9$ years $^2$ Because we used the population mean $({\mu})$ to compute the squared deviations from the mean , we can find the variance $({\sigma^2})$ , without introducing any bias, by simply averaging the squared deviations from the mean $ {\sigma^2} = \dfrac{\sum\limits_{i=1}^{{N}} (x_i - {\mu})^2}{{N}} $ $ {\sigma^2} = \dfrac{{64} + {25} + {100} + {9}} {{4}} $ $ {\sigma^2} = \dfrac{{198}}{{4}} = {49.5\text{ years}^2} $ As you might guess from the notation, the population standard deviation $({\sigma})$ is found by taking the square root of the population variance $({\sigma^2})$ ${\sigma} = \sqrt{{\sigma^2}}$ $ {\sigma} = \sqrt{{49.5\text{ years}^2}} = {7\text{ years}} $ The average sloth at the zoo is 10 years old. There is a standard deviation of 7 years.